[br]Move the sliders for [math]m[/math]and [math]c[/math] to explore the properties of a straight line graph, otherwise known as a [b]LINEAR GRAPH[/b].[b][br][br]Question 1: [br][/b][br] What feature of the [color=#ff0000]red line[/color] does [math]c[/math] represent?[br][br][b]Question 2:[br][/b][br] [b] a)[/b] What do you notice about the [color=#ff0000]red line[/color] as [math]m[/math] gets larger (greater than 1)?[br][br][b] b) [/b]What do you notice about the [color=#ff0000]red line[/color] as [math]m[/math] gets smaller (between 0 and 1)?[br][br][b] c) [/b]What do you notice when [math]m[/math] is negative?[br][br] [b] d) [/b]What do you notice when [math]m=0[/math]?[br][br][b] e) [/b]What feature of the [color=#ff0000]red line[/color] does [math]m[/math] represent?
[b]Question 3:[/b] [br][br] Using your investigation from Questions 1 and 2, move the sliders for [math]m[/math] and [math]c[/math] to determine the equation of [br] at least 2 random lines.[br][i][br](Feel free to find the equations of more than just 2 lines. Practice makes perfect!)[/i] [br]
[br]In Mathematics we have a special term to refer to the ‘steepness’ / ‘slope’ of a line. We call it the [b]GRADIENT .[/b][br][br]The [b]gradient [/b]of a straight line is denoted by the symbol[size=100][size=150] [math]m[/math] [/size][/size]in the linear equation: [math]y=mx+c[/math]. [br][br]The definition of a gradient is the: [br][br][center]“[i]Change in vertical distance [math](y)[/math] [/i][i]as the horizontal distance [math](x)[/math] [/i][i]increases by 1 unit”.[/i][/center]However, we more often use the definition/formula:[br][br][center][math]m=\frac{rise}{run}[/math] [/center]Lets investigate...[b][br][br]Question 4:[br][br] [/b]Calculate the gradient between the points:[br][br][b] a) [/b](0,0) and (1,2) [b]b) [/b](0,0) and (2,2) [b]c) [/b](0,0) and (2,1)[b] d) [/b](0,0) and (3,-3) [b]d) [/b](0,0) and (2,3)[br][br] (Hint: Change the value of [math]run[/math] first.)[br][b][br] e) [/b]When finding the gradient of a linear graph, which points can be used to calculate [math]m=\frac{rise}{run}[/math]?[br][br] (Hint: The [color=#0000ff]blue [/color]dot on the can be dragged up and down your [color=#ff0000]line [/color]using your cursor.)[b][br][br]Question 5: [br][br][/b] [b]a) [/b]In the equation of a straight line, the gradient ([math]m[/math]) is the coefficient of ___ [br][br][b] b) [/b]Describe how you would determine the gradient of a straight line from its graph?[br][br][b] c) [/b]Can the [math]run[/math] between two points be negative? Can the [math]rise[/math] between two points be negative?[br][br][b] d)[/b] What must the gradient be if the rise between two points is zero ([math]rise=0[/math])? [br][br][b] e) [/b]How do you calculate the gradient of a straight line when [i]only given the coordinates of two points[/i]?
[br]Put your new skills to the test! Try to determine the equation of the following [u]9 lines![/u] [br][br][b]Question 6: [/b][br][br] Determine the equations of the lines [b]a)[/b] though [b]i). [br][br][/b] Check that each of your equations are correct by typing them in and seeing if you get a [color=#6aa84f]match[/color].[br][br][i](Two blue points have been given in the top-left which can be moved to help visualize your calculation of 'rise' and 'run')[/i]
[br]Any two points can form a a straight line. Let's investigate how we use this to sketch [b]horizontal[/b] and [b]vertical lines[/b][br][b][br]Question 7:[/b][br][br] Try to make vertical and horizontal lines:[br][br] [b]a) [/b][math]y=4[/math] [b]b)[/b] [math]y=-2[/math] [b]c) [/b][math]y=0[/math][br] [br] [b]d) [/b][math]x=1[/math] [b]e) [/b][math]x=-2.5[/math] [b]f) [/b][math]x=0[/math][br][b][br]Question 8:[br][br][/b] [b]a) [/b]How do you determine the equation of a vertical line given its linear graph?[br][b][br] b) [/b]How do you determine the equation of a horizontal line given its linear graph?
[br]You may have noticed that linear graphs can be very long (in fact they are infinitely long!). So what must we do if we only want a segment of a line? ... We use [b]LINE SEGMENTS.[br][br][/b]Below you can sketch a [color=#ff0000]sloped / horizontal line segment [/color]and a [color=#0000ff]vertical line segment.[br][br][/color]Try entering your straight line equations into the input boxes (just as with Q6), but this time include maximum and minimum values for [math]x[/math]and [math]y[/math]![br][br][i]Be sure to see what happens if you 'tick' the "Show Max/Min Lines" box during your investigation.[/i][b][br][br]Question 9:[br][br][/b] [b]a) [/b]What does the 'domain' of a linear graph represent?[br][br] [b]b)[/b] What does the 'range' of a linear graph represent?[br][b][br] c) [/b]Can you think of mnemonic device that which help you remember the differences and/or similarities of [br] 'domain' and 'range'.[b][br][br]Question 10:[br][br][/b] [b]a) [/b]Sketch a [color=#ff0000]positively sloped[/color] [color=#ff0000]line segment[/color] and [color=#0000ff]vertical line segment[/color] that join at end points.[br][br][b] b) [/b]Sketch a [color=#ff0000]negatively sloped[/color] [color=#ff0000]line segment[/color] and [color=#0000ff]vertical line segment[/color] that join at end points.[br] [b][br] c) [/b]Sketch a [color=#ff0000]horizontal [/color][color=#ff0000]line segment[/color] and [color=#0000ff]vertical line segment[/color] that join at end points.[br]